Problem: All of the 5th grade teachers and students from Covington went on a field trip to an archaeology museum. Tickets were $$8.00$ each for teachers and $$2.50$ each for students, and the group paid $$41.00$ in total. The next month, the same group visited a natural history museum where the tickets cost $$32.00$ each for teachers and $$12.50$ each for students, and the group paid $$189.00$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8x+2.5y = 41}$ ${32x+12.5y = 189}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-32x-10y = -164}$ ${32x+12.5y = 189}$ Add the top and bottom equations together. $ 2.5y = 25 $ $ y = \dfrac{25}{2.5}$ ${y = 10}$ Now that you know ${y = 10}$ , plug it back into $ {8x+2.5y = 41}$ to find $x$ ${8x + 2.5}{(10)}{= 41}$ $8x+25 = 41$ $8x = 16$ $x = \dfrac{16}{8}$ ${x = 2}$ You can also plug ${y = 10}$ into $ {32x+12.5y = 189}$ and get the same answer for $x$ ${32x + 12.5}{(10)}{= 189}$ ${x = 2}$ There were $2$ teachers and $10$ students on the field trips.